Traditional multi-grid method is a way of efficiently solving a largesystem of algebraic equations, which may arise from the discretizationof some partial differential equations. For this reason, theeffective operators used at each level can all be regarded as anapproximation to the original operator at that level. In recentyears, Brandt has proposed to extend the multi-grid method to caseswhen the effective problems solved at different levels correspond tovery different kinds of multi-scale analysis models (Brandt, 2002). For example, themodels used at the finest level might be molecular dynamics or MonteCarlo models whereas the effective models used at the coarse levelscorrespond to some continuum models. Brandt noted that there is noneed to have closed form macroscopic models at the coarse scale sincecoupling to the models used at the fine scale grids automaticallyprovides effective models at the coarse scale. Brandt also noted thatone might be able to exploit scale separation to improve theefficiency of the algorithm, by restricting the smoothing operationsat fine grid levels to small windows and for few sweeps.
Multiscale Analysis of Advanced Materials
- But forcomplex fluids, this would result in rather different kinds of models.This is one of the starting points of multiscale modeling.
- Proposed pipelines are often tailored to individual studies, leading to a fragmented landscape of available methods, and no clear guidance about which statistical tools are best suited to a particular question.
- The key is that the user must be very aware of the assumptions and bounds of their model when employing one of these techniques.
- These slowly varying quantities aretypically the Goldstone modes of the system.
The latter puts constraints on the approximate solution, which are called solvability conditions. The renormalization group method is one of the most powerfultechniques for studying the effective behavior of a complex system inthe space of scales (Wilson and Kogut, 1974). The basic object of interest is adynamical system for the effective model in which the time parameteris replaced by scale. Therefore this dynamical system describes howthe effective model changes as the scale changes. Partly forthis reason, the same approach has been followed in modeling complexfluids, such as polymeric fluids.
Methods
The firstis that the implementation of CPMD is based on an extended Lagrangianframework by considering the wavefunctions for electrons in the samesetting as the positions of the nuclei. In this extended phase space,one can write down a Lagrangian which incorporates both theHamiltonian for the nuclei and the wavefunctions. The second is the choice of the massparameter for the wavefunctions. Perhaps the most natural choiceshould be the mass of an electron. This makes the system stiffsince the time scales of the electrons and the nuclei are quitedisparate. However, since we are only interested in the dynamics ofthe nuclei, not the electrons, we can choose a value which is muchlarger than the electron mass, so long as it still gives ussatisfactory accuracy for the nuclear dynamics.
Motivation for multiple-scale analysis
Solving each scale individually and linking their results is much faster than trying to solve a single high-resolution model containing all relevant details. Modelingadvanced materials accurately is extremely complex because of the high numberof variables at play. The materials in question are heterogeneous in nature,meaning they have more than one pure constituent, e.g. carbon fiber + polymerresin or sedimentary rock + gaseous pores. The first scheme to address this problem is what VanDyke (1975) refers to as the method of strained coordinates.The method is sometimes attributed to Poincare, although Poincarecredits the basic idea to the astronomer Lindstedt(Kevorkian and Cole, 1996). Lighthill introduced a more general version in 1949.Later Krylov and Bogoliubov and Kevorkian and Cole introduced thetwo-scale expansion, which is now the more standard approach. Vanden-Eijnden, “A computational strategy for multiscale chaotic systems with applications to Lorenz 96 model,” preprint.
The need for multiscale modeling comes usually from the fact that theavailable macroscale models are not accurate enough, and themicroscale models are not efficient enough and/or offer too muchinformation. By combining both viewpoints, one hopes to arrive at areasonable compromise between accuracy and efficiency. The most efficient solution is to use multiscale FEA to divide and conquer the problem. To accomplish this, a local scale model of the material microstructure is embedded within the global scale FE model of the part. Then, the software analyzes the different scales simultaneously.
- In addition, the computational benefits afforded by concurrently coupling the micro- and macroscale models are also presented.
- HMM has been used on a variety of problems, includingstochastic simulation algorithms (SSA) with disparate rates,elliptic partial differential equations with multiscale data,and ordinary differential equations (ODE) with multiple time scales.
- In sequential multiscalemodeling, one has a macroscale model in which some details of theconstitutive relations are precomputed using microscale models.
- Future studies should seek to experimentally validate the multiscale geometries presented within this paper.
- The hope is that by using such amulti-scale (and multi-physics) approach, one might be able to strikea balance between accuracy (which favors using more detailed andmicroscopic models) and feasibility (which favors using less detailed,more macroscopic models).
Examples of multiscale methods
The first problem is that simplicity is largely lost.In order to model the complex rheological properties of polymer fluids,one is forced to make more complicated constitutive assumptions withmore and more parameters. For polymer fluids we are often interested inunderstanding how the conformation of the polymer interacts with theflow. This kind of information is missing in the kind of empiricalapproach described above. Macroscale models require constitutive relations which are almost always obtained empirically, by guessing.
In each case, excellent agreement is noted between the high-fidelity simulations and the corresponding optimized macroscale displacement fields, with errors of less than 10% noted in all instances. Significantly, strains exceeding 18%, 20%, and 22% are observed in each respective case, which considering the excellent agreement with high-fidelity simulations, demonstrates this framework’s capability to function effectively in the nonlinear elastic regime. The necessity of nonlinear analysis at both scales is verified by comparing the results of the high-fidelity simulations against identical simulations performed using linear elasticity. In all instances, the error in displacement significantly increases, with some over an order of magnitude larger.
We refer to thefirst type as type A problems and the second type as type B problems. In the multiscale approach, one uses a variety of models at differentlevels of resolution and complexity to study one system. Thedifferent models are linked together either analytically ornumerically. For example, one may study the mechanical behavior ofsolids using both the atomistic and continuum models at the same time,with the constitutive relations needed in the continuum model computedfrom the atomistic model. The hope is that by using such amulti-scale (and multi-physics) approach, one might be able to strikea balance between accuracy (which favors using more detailed andmicroscopic models) and feasibility (which favors using less detailed,more macroscopic models).
